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Let’s Investigate

Let’s Investigate. The Tangent Ratio. The Tangent Angle. The Sine Ratio. The Sine Angle. The Cosine Ratio. The Cosine Angle. Mixed Problems. Extension. Starter Questions. www.mathsrevision.com. Trigonometry. Let’s Investigate!. Trigonometry means “triangle” and “measurement”.

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Let’s Investigate

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  1. Let’s Investigate The Tangent Ratio The Tangent Angle The Sine Ratio The Sine Angle The Cosine Ratio The Cosine Angle Mixed Problems Extension

  2. Starter Questions www.mathsrevision.com www.mathsrevision.com

  3. Trigonometry Let’s Investigate! www.mathsrevision.com

  4. Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite hypotenuse x° Adjacent

  5. Mathemagic! Opposite hypotenuse 30° Adjacent Opposite = 0.6 Adjacent

  6. Try another! Opposite hypotenuse 45° Adjacent Opposite = 1 Adjacent

  7. Opposite 0.6 = Adjacent Opposite is called the tangent of an angle. Adjacent For an angle of 30°, We write tan 30° = 0.6

  8. The ancient Greeks discovered this and repeated this for possible angles. Tan 30° = 0.577 Accurate to 3 decimal places!

  9. Now-a-days we can use calculators instead of tables to find the Tan of an angle. On your calculator press Tan Followed by 30, and press = Notice that your calculator is incredibly accurate!! Accurate to 9 decimal places!

  10. What’s the point of all this??? Don’t worry, you’re about to find out!

  11. Opp How high is the tower? 60° 12 m

  12. Copy this! Opposite hypotenuse 60° 12 m Adjacent

  13. Copy this! Change side, change sign! Opp Tan x° = Adj Opp Tan 60° = 12 12 x Tan 60° = Opp Opp = 12 x Tan 60° = 20.8m (1 d.p.)

  14. 20.8m Don’t worry, you’ll be trying plenty of examples!! ? So the tower’s 20.8 m high!

  15. Starter Questions 3cm www.mathsrevision.com www.mathsrevision.com

  16. Opp Tan x° = Adj Opposite x° Adjacent

  17. Change side, change sign! Example Opp Hyp Opp c Tan x° = Adj 65° c Tan 65° = 8m 8 Adj 8 x Tan 65° = c c = 8 x Tan 65° = 17.2m (1 d.p.)

  18. Now try Exercise 1. (HSDU Support Materials)

  19. Starter Questions www.mathsrevision.com www.mathsrevision.com

  20. Using Tan to calculate angles www.mathsrevision.com

  21. Example SOH CAH TOA Opp Hyp 18m Opp ? Tan x° = Adj x° 12m 18 Tan x° = Adj 12 Tan x° = 1.5

  22. Tan x° = 1.5 We need to use Tan ⁻¹on the calculator. Tan ⁻¹ Tan How do we find x°? Tan ⁻¹is written above Followed by Tan To get this press 2nd

  23. Tan x° = 1.5 Tan ⁻¹ Tan 2nd Press Enter = 1.5 Tan ⁻¹1.5 x = = 56.3° (1 d.p.)

  24. Now try Exercise 2. (HSDU Support Materials)

  25. Starter Questions www.mathsrevision.com www.mathsrevision.com

  26. The Sine Ratio Opp Sin x° = Hyp Opposite hypotenuse x°

  27. Change side, change sign! Example Hyp 11cm O Opp Opp Sin x° = 34° Hyp O Sin 34° = 11 = O 11 x Sin 34° O = 11 x Sin 34° = 6.2cm (1 d.p.)

  28. Now try Exercise 3. (HSDU Support Materials)

  29. Starter Questions www.mathsrevision.com 57o www.mathsrevision.com

  30. Using Sin to calculate angles www.mathsrevision.com

  31. Example Hyp 9m 6m SOH CAH TOA Opp x° Opp ? Sin x° = Hyp 6 Sin x° = 9 Sin x° = 0.667 (3 d.p.)

  32. Sin x° =0.667 (3 d.p.) We need to use Sin ⁻¹on the calculator. Sin ⁻¹ Sin How do we find x°? Sin ⁻¹is written above Followed by Sin To get this press 2nd

  33. Sin x° = 0.667 (3 d.p.) Sin ⁻¹ Sin Press 2nd Enter 0.667 = x = Sin ⁻¹0.667 = 41.8° (1 d.p.)

  34. Now try Exercise 4. (HSDU Support Materials)

  35. Starter Questions www.mathsrevision.com www.mathsrevision.com

  36. The Cosine Ratio Adj Cos x° = Hyp hypotenuse x° Adjacent

  37. Change side, change sign! Example b Adj 40° Adj Cos x° = Opp Hyp Hyp 35mm b Cos 40° = 35 35 x Cos 40° = b b = 35 x Cos 40° = 26.8mm (1 d.p.)

  38. Now try Exercise 5. (HSDU Support Materials)

  39. Starter Questions Q1. Calculate Q2. Round to 1 decimal place 2.354. Q3. How many minutes in 3hours www.mathsrevision.com Q4. The answer to the question is 180. What is the question. www.mathsrevision.com

  40. Using Cos to calculate angles www.mathsrevision.com

  41. Example SOH CAH TOA Adj 34cm x° Adj Cos x° = Opp Hyp Hyp 45cm 34 Cos x° = 45 Cos x° = 0.756 (3 d.p.) x = Cos ⁻¹0.756 =40.9° (1 d.p.)

  42. Now try Exercise 6. (HSDU Support Materials)

  43. Starter Questions www.mathsrevision.com www.mathsrevision.com

  44. The Three Ratios Sine Tangent Cosine Sine Sine Tangent www.mathsrevision.com Cosine Cosine Sine www.mathsrevision.com

  45. Opp Adj Opp Sin x° = Cos x° = Tan x° = Hyp Hyp Adj The Three Ratios

  46. Opp Adj Opp Copy this! Sin x° = Cos x° = Tan x° = Hyp Hyp Adj O S H A C H O T A SOH CAH TOA

  47. Mixed Examples Cos 20° Tan 27° Sin 36° Sin 60° Sin 30° Tan 40° www.mathsrevision.com Cos 12° Cos 79° Sin 35° www.mathsrevision.com

  48. Change side, change sign! Example 1 SOH CAH TOA Hyp 15m O Opp Opp Sin x° = 40° Hyp O Sin 40° = 15 15 x Sin 40° = O O = 15 x Sin 40° = 9.6m (1 d.p.)

  49. Change side, change sign! Example 2 SOH CAH TOA b Adj 35° Adj Opp Cos x° = Hyp Hyp 23cm b Cos 35° = 23 23 x Cos 35° = b b = 23 x Cos 35° = 18.8cm (1 d.p.)

  50. Change side, change sign! Example 3 SOH CAH TOA Opp Hyp c Opp Tan x° = Adj 60° c 15m Tan 60° = 15 Adj 15 x Tan 60° = c c = 15 x Tan 60° = 26.0m (1 d.p.)

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